Indeed, for \(r>0\)  let \(Q:=x_0+(-r,r)^n\subset U\)  be any cube centered at \(x\in U\)  and let \(0<\theta ^{\prime \prime }<\theta ^{\prime }<1.\)  For shortness, write \(Q^{\prime }:=x_0+(-\theta ^{\prime } r,\theta ^{\prime }r)^n\)  and \(Q^{\prime \prime }:=x_0+(-\theta ^{\prime \prime }r,\theta ^{\prime \prime }r)^n.\)  By [1]} (see also [2]}) there exists a \(\mathcal {L}^2\) -measurable set \(\omega \subset Q^{\prime }\)  and a rigid displacement \(a:\mathbb {R}^2\rightarrow \mathbb {R}^2\)  such that \(|\omega |\le c_* r\mathcal {H}^{n-1}(J_u)=0\)  and
\(\int _{Q^{\prime }} |u -a|^{\frac{2n}{n-1}}dx = \int _{Q^{\prime }\setminus \omega } |u -a|^{\frac{2n}{n-1}}dx\le c_*r^2 \Big (\int _Q |e(u)|^2\Big )^{\frac{n}{n-1}},\) 
where \(c_*\)  depends only on \(n.\)  Hence, \(u\in L_\mathrm {loc}^{\frac{2n}{n-1}}(Q).\)  Next, fix any mollifier \(\rho _1\in C^\infty (B_r(0))\)  with \(\rho _\epsilon \in C_c^\infty (B_{(\theta ^{\prime \prime } - \theta ^{\prime })\epsilon }),\)  where \(\rho _\epsilon (x):=\rho _1(x/\epsilon ),\)  \(\epsilon \in (0,r).\) 
By [1]} there exists \(\bar{p}>0\)  depending on \(n\)  and \(\epsilon \)  such that
\(\int _{Q^{\prime \prime }} |e(u*\rho _\epsilon ) - e(u)*\rho _\epsilon |^2 dx \le c \Big (\frac{\mathcal {H}^{n-1}(J_u)}{r^{n-1}}\Big )^{\bar{p}} \int _Q |e(u)|^2dx =0,\) 
where \(c\)  depends on \(n,\)  \(\rho _1\)  and \(\epsilon .\) 
Hence,
\(e(u*\rho _\epsilon ) = e(u)*\rho _\epsilon \quad \text{a.e. in $Q^{\prime \prime }.$}\) 
Recall that \(u*\rho _\epsilon \in C^\infty (Q^{\prime \prime }).\)  Since \(e(u)\in L^2(Q),\)  \(e(u)*\rho _\epsilon \in C^\infty (Q^{\prime \prime })\cap L^2(Q^{\prime \prime })\)  in particular, \(e(u*\rho _\epsilon ) \in C^\infty (Q^{\prime \prime })\cap L^2(Q^{\prime \prime }).\)  By Poincaré-Korn inequality \(u*\rho _\epsilon \in H^1(Q^{\prime \prime }).\)  Since \(e(u)*\rho _\epsilon \rightarrow e(u)\)  in \(L^2(Q^{\prime \prime })\)  as \(\epsilon \rightarrow 0,\)  in view of (REF ) there exists \(\epsilon _0>0\)  such that
\(\Vert e(u*\rho _\epsilon )\Vert _{L^2(Q^{\prime \prime })} \le \Vert e(u)\Vert _{L^2(Q^{\prime \prime })} +1\quad \text{for all $\epsilon \in (0,\epsilon _0).$}\) 
Moreover, by Poincaré-Korn inequality for any \(\epsilon \in (0,\epsilon _0)\)  there exists a rigid displacement \(a_\epsilon \)  such that
\(\Vert  u*\rho _\epsilon - a_\epsilon \Vert _{H^1(Q^{\prime \prime })} \le C \Vert e(u*\rho _\epsilon )\Vert _{L^2(Q^{\prime \prime })} \le C(\Vert e(u)\Vert _{L^2(Q^{\prime \prime })} + 1),\) 
where \(C\)  is the Poincaré-Korn constant for a cube.
Thus, the family \(\lbrace u*\rho _\epsilon \rbrace _\epsilon \)  is uniformly bounded in \(H^1(Q^{\prime \prime })\) . Since \(u*\rho _\epsilon \rightarrow u\)  in \(L^2(Q^{\prime \prime }),\)  there exists a rigid displacement \(a\)  such that \(a_\epsilon \rightarrow a\)  in \(L^2(Q^{\prime \prime }).\)  Then \(u*\rho _\epsilon - a_\epsilon \)  weakly converges to \(u-a\)  in \(H^1(Q^{\prime \prime }),\)  i.e., \(u-a\in H^1(Q^{\prime \prime }).\)  Since \(a\)  is linear and \(\theta ^{\prime \prime }\)  is arbitrary, \(u\in H_\mathrm {loc}^1(Q).\)  Now covering \(U\)  with finitely many cubes of edgelength \(2r\)  we get \(u\in H_\mathrm {loc}^1(U).\)
